No, you don't have a lot of choice. UK banks have many restrictions about who can open an account. Just by queuing in a branch I've seen immigrants being rejected because they don't have a job yet.
If you still think about banks re-loaning bank deposits you're living in another century. Fractional reserve banking hasn't been a thing since (at least) the 1960s.
Depositing money isn't giving money to a bank. It's exchanging money for highly liquid debt and receiving interest in return. Giving money to a bank is a relationship where the bank pays you.
Also, the government already forces you to have auto insurance. This doesn't really seem any better or worse.
You have a huge variety of choice of bank, including nonprofit, cooperative credit unions that don't do anything evil to their customers.